Considered community consists of $N$ agents entitled to take part in the decision process. Each of them is endowed with the income $y_i$.
Let's assume that $i$th agent has preferences over private consumption $c_i$ and level of the provision of public good $P_g\cdot G = \displaystyle\sum_{j=1}^k p_j$:
\begin{equation}
u^i(c_i,P_g)=c_i+\alpha_i\cdot P_G\cdot G
\end{equation}
where $k$ is a number of different public goods that may be provided (in our example $k=3$). For simplification private consumption is aggregated in a singleton $c_i$. $\alpha_i$ is the parameter which tells how much the unit of public good is valued more than the unit of private consumption. 

%Every public good has the price $p_j$ dependent on the level $l_j\in [0,1]$ of its provision:
%\begin{equation}
%p_j = P_j\cdot l_j
%\end{equation}

The public goods are provided by agents' contributions $x_i$:
\begin{equation}
P_g\cdot G = \sum_{i=1}^n x_i = x_i + x_{-i}
\end{equation}
where $x_{-i} = \sum_{m\ne i} x_m$.

$i$th agent considers his own budget as:
\begin{equation}
y_i = c_i + x_i
\end{equation}


